Exercise 19, § 2

In a metric space every relatively compact set is bounded. If $X$ is a topological space, show that the following conditions on $X$ are equivalent: $\alpha$ there exists a metric on $X$, compatible with its topology, such that every bounded subset of $X$ (with respect to this metric) is relatively compact; $\beta$ $X$ is locally compact and has a countable base. [To show that $\alpha) \Longrightarrow \beta$), show that if every bounded set is relatively compact, then $X$ is locally compact and $\sigma$-compact. To show that $\beta) \Longrightarrow \alpha$), note that $X$ is metrizable by the corollary to Proposition 12; if $d$ is a metric on $X$ compatible with the topology of $X$, and if $f$ is the function defined in § 1, Exercise 15 d), take the uniformity on $X$ defined by the two pseudometrics $d(x, y)$ and $|f(x) - f(y)|$.]